358 CONDORCET. 



Then (f> (q) -^'^ (q) is the probability that there will be a valid 

 decision, <^ (q) is the probability that there will be a valid and 

 correct decision, and yjr (q) is the probability that there will be a 

 valid and incorrect decision. Moreover 1 — -yfr (q) is the probability 

 that there will not be an incorrect decision, and 1 — cj) {q) is the 

 probability that there will not be a correct decision. 



It will be observed that here (q) + yjr (q) is not equal to unity. 

 In fact 1 — <f> {q) — yjr (q) consists of all the terms in the expansion 

 of (v + e)'^^^^ lying between those which involve v^'^'^'^^ e'^'^' and 

 ^«-<z' g3+2'+i both exclusive. Thus 1 — cj) (q) — ylr (q) is the probability 

 that the decision will be invalid for want of the prescribed 

 plurality. 



It is shewn by Condorcet that if v is greater than e the 

 limit of <p {q) when q increases indefinitely is unity. See his 

 pages 19 — 21. 



670. Suppose we know that a valid decision has been given, 

 but do not know the numbers for and against. Then the pro- 

 bability that the decisian is correct is , , , . , ^ , and the pro- 



bability that it is incorrect is r ^^ 



<l>iq)+'f{q)' 



Suppose we know that a valid decision has been given, and 

 also know the numbers for and against. Then the probabilities 

 of the correctness and incorrectness of the decision are those which 

 have been stated in Art. 667. 



671. We will now indicate what Condorcet appears to mean 

 by the principal conditions which ought to be secured in a de- 

 cision ; they are : 



1. That an incorrect decision shall not be given ; that is 

 l — '^iq) must be large. 



2. That a correct decision shall be given ; that is <p (q) must 

 be large. 



3. That there shall be a valid decision, correct or incorrect ; 

 that is </) (2') + '^ (q) must be large. 



4. That a valid decision which has been given is correct, 



